A class of generalized shift-splitting preconditioners for nonsymmetric saddle point problems

where h =1/( p +1) denotes the discretization mesh size. This kind of systems of linear equations with two‐by‐two block structure arises in many applications, such as discrete Stokes equations, shift‐splitting iteration scheme for solving nonsymmetric saddle‐point problems, and image reconstruction and restoration …”

Section: Numerical Examplesmentioning

confidence: 99%

“…where h = 1∕(p + 1) denotes the discretization mesh size. This kind of systems of linear equations with two-by-two block structure arises in many applications, such as discrete Stokes equations, 29 shift-splitting iteration scheme for solving nonsymmetric saddle-point problems, 34 and image reconstruction and restoration. 35 Moreover, this example can be considered as a generalization of example 4.1 in the work of Bai et al 10 In the computation, we choose = 1∕2.…”

Section: Preconditioner Descriptionmentioning

confidence: 99%

Scaled norm minimization method for computing the parameters of the HSS and the two‐parameter HSS preconditioners

Yang

2018

Numerical Linear Algebra App

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Summary The performance of the Hermitian and skew‐Hermitian splitting (HSS) preconditioner for the non‐Hermitian positive definite system of linear equations is largely dependent on the choice of its parameter value. In this work, an efficient scaled norm minimization (SNM) method is proposed to compute the parameter value of the HSS preconditioner. In addition, by choosing different parameters for the Hermitian and the skew‐Hermitian matrices in the HSS preconditioner, a two‐parameter HSS preconditioner is proposed. Moreover, an efficient and practical formula for computing the parameter values of this new preconditioner is also derived by using the SNM method. Numerical examples are illustrated to verify the performances of the HSS and the two‐parameter HSS preconditioners when their parameters are computed by the SNM method.

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“…In this section, inspired by the ideas of [20,18,33], we develop a new splitting called the modified generalized shift-splitting (MGSSP) of the nonsymmetric saddle point matrix A by combining the generalized splitting-splitting and the modified shift-splitting of the saddle point matrix A as follows.…”

Section: The Modified Generalized Shift-splitting (Mgssp) Preconditiomentioning

confidence: 99%

“…Besides, Bai et al put forward the well-known Hermitian and skew-Hermitian splitting (HSS) methods [7] and its variants [6,8,9,5]. On the basis of the shift-splitting (SS) of a non-Hermitian matrix [12], Cao et al [17] derived the SS iteration method as well as the SS preconditioner for the nonsingular saddle point problems, and Chen and Ma [20] and Cao et al [18] generalized the SS iteration method and obtained the generalized SS (GSS) iteration method. To increase the convergence rate of the GSS iteration method, Huang and Su [33] newly developed the modified shift-splitting (MSSP) iteration method.…”

Section: Introductionmentioning

confidence: 99%

A modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems

et al. 2017

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For the nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) parts, the modified generalized shift-splitting (MGSSP) preconditioner as well as the MGSSP iteration method are derived in this paper, which generalize the MSSP preconditioner and the MSSP iteration method newly developed by Huang and Su (J. Comput. Appl. Math. 2017), respectively. The convergent and semi-convergent analysis of the MGSSP iteration method are presented, and we prove that this method is unconditionally convergent and semi-convergent. In addition, some spectral properties of the preconditioned matrix are carefully analyzed. Numerical results demonstrate the robustness and effectiveness of the MGSSP preconditioner and the MGSSP iteration method, and also illustrate that the MGSSP iteration method outperforms the GSS and GMSS iteration methods, and the MGSSP preconditioner is superior to the shift-splitting (SS), generalized SS (GSS), modified SS (MSS) and generalized MSS (GMSS) preconditioners for the GMRES method for solving the nonsymmetric saddle point problems.for the saddle point problem (1), where α is a positive constant and I is the identity matrix.where α ≥ 0, β > 0 and I is the identity matrix. It is easy to see that P SS is a special case of P GSS when α = β. Numerical results in [21,20] confirmed that the GSS preconditioner is superior to the SS preconditioner.

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A class of generalized shift-splitting preconditioners for nonsymmetric saddle point problems

Cited by 59 publications

References 16 publications

An extension of the positive-definite and skew-Hermitian splitting method for preconditioning of generalized saddle point problems

An extension of the positive-definite and skew-Hermitian splitting method for preconditioning of generalized saddle point problems

Scaled norm minimization method for computing the parameters of the HSS and the two‐parameter HSS preconditioners

A modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems

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